Because this is such a sensitive topic, you have to be careful not to confuse what is meant by race with ancestry, nationality, culture, religion, or even ethnicity. Unfortunately this is quite complicated, and can vary between different contexts.
To give you some idea of how complicated it is, here is a good discussion of the difference between race, ethnicity, nationality, culture, and heritage, and from a pamphlet published by the US Census Bureau. Note the definitions from these two sources don't match. The first says "race" and "ethnicity" are mostly the same, while the census bureau has a much more narrow definition of "ethnicity".
To add to this complexity, what might be considered inoffensive in one context might be thought bigoted in another. For example, you use the term "mixed-race" to describe someone who is the child of an American and a German. Many would not consider either to be an example of "race" at all, but simply nationality. There are, after all, many Americans whose ancestors were from Germany. Your misuse of the term might be considered an example of racism, or at least ignorance.
For this reason, it may be better to avoid the topic entirely. It will be difficult to avoid offense if you are not even aware of what kind of statements might cause offense. Here is a funny (and not-so-funny) video that illustrates the kind of well-meaning but offensive behavior some people encounter on a regular basis: What kind of Asian are you?
With all that out of the way, "heritage" is a word you can use which (at least at the moment) should be pretty safe. Heritage is a generally positive-sounding term that refers to all aspects of a person's ancestry. Someone whose parents are from different countries and someone whose parents are of different races can both be said to be of mixed-heritage.
(Note: "Mixed-race" is a little more problematic, as it may cause offense in the wrong context. It's also difficult to define -- what percentage of your overall racial makeup needs to be from a certain race in order to claim that race as part of your heritage? It's not an abstract question, as the answer can mean certain types of preferential treatment or government support, or even political repercussions.)
Note this is only safe in the context of talking about someone's ancestry. It's not a good idea to characterize someone by their heritage, because that can lead to accusations of offensive stereotyping. For example, the woman in the video might say she is of Korean heritage, but her nationality is American. Her life may include aspects of both cultures, but that's her choice and -- as you can see in the video -- one shouldn't make assumptions.
Side note: When talking strictly about nationality, someone who holds passports from two (or more) different countries is said to have multiple/dual citizenship, or multiple/dual nationality**.